# Theory Wfrec

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theory Wfrec
imports Main
`(*  Title:      HOL/Library/Wfrec.thy    Author:     Tobias Nipkow    Author:     Lawrence C Paulson    Author:     Konrad Slind*)header {* Well-Founded Recursion Combinator *}theory Wfrecimports Mainbegininductive  wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"  for R :: "('a * 'a) set"  and F :: "('a => 'b) => 'a => 'b"where  wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>            wfrec_rel R F x (F g x)"definition  cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b" where  "cut f r x == (%y. if (y,x):r then f y else undefined)"definition  adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool" where  "adm_wf R F == ALL f g x.     (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"definition  wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where  "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"by (simp add: fun_eq_iff cut_def)lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"by (simp add: cut_def)text{*Inductive characterization of wfrec combinator; for details see:John Harrison, "Inductive definitions: automation and application"*}lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"apply (simp add: adm_wf_def)apply (erule_tac a=x in wf_induct)apply (rule ex1I)apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)apply (fast dest!: theI')apply (erule wfrec_rel.cases, simp)apply (erule allE, erule allE, erule allE, erule mp)apply (fast intro: the_equality [symmetric])donelemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"apply (simp add: adm_wf_def)apply (intro strip)apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)apply (rule refl)donelemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"apply (simp add: wfrec_def)apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)apply (rule wfrec_rel.wfrecI)apply (intro strip)apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])donetext{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"apply autoapply (blast intro: wfrec)donesubsection {* Nitpick setup *}axiomatization wf_wfrec :: "('a × 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"definition wf_wfrec' :: "('a × 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where[nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"definition wfrec' ::  "('a × 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where"wfrec' R F x ≡ if wf R then wf_wfrec' R F x                else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"setup {*  Nitpick_HOL.register_ersatz_global    [(@{const_name wf_wfrec}, @{const_name wf_wfrec'}),     (@{const_name wfrec}, @{const_name wfrec'})]*}hide_const (open) wf_wfrec wf_wfrec' wfrec'hide_fact (open) wf_wfrec'_def wfrec'_defsubsection {* Wellfoundedness of @{text same_fst} *}definition same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"where    "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"   --{*For @{text rec_def} declarations where the first n parameters       stay unchanged in the recursive call. *}lemma same_fstI [intro!]:     "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"by (simp add: same_fst_def)lemma wf_same_fst:  assumes prem: "(!!x. P x ==> wf(R x))"  shows "wf(same_fst P R)"apply (simp cong del: imp_cong add: wf_def same_fst_def)apply (intro strip)apply (rename_tac a b)apply (case_tac "wf (R a)") apply (erule_tac a = b in wf_induct, blast)apply (blast intro: prem)doneend`